Cesàro Convergent Sequences in the Mackey Topology

Abstract

A Banach space X is said to have property (\(\mu ^s\)) if every weak\(^*\)-null sequence in \(X^*\) admits a subsequence, such that all of its subsequences are Cesaro convergent to 0 with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapien. We prove that property \((\mu ^s)\) holds for every subspace of a Banach space which is strongly generated by an operator with Banach–Saks adjoint (e.g., a strongly super weakly compactly generated space). The stability of property \((\mu ^s)\) under \(\ell ^p\)-sums is discussed. For a family \(\mathcal {A}\) of relatively weakly compact subsets of X, we consider the weaker property \((\mu _\mathcal {A}^s)\) which only requires uniform convergence on the elements of \(\mathcal {A}\), and we give some applications to Banach lattices and Lebesgue–Bochner spaces. We show that every Banach lattice with order continuous norm and weak unit has property \((\mu _\mathcal {A}^s)\) for the family of all L-weakly compact sets. This sharpens a result of de Pagter, Dodds, and Sukochev. On the other hand, we prove that \(L^1(\nu ,X)\) (for a finite measure \(\nu \)) has property \((\mu _\mathcal {A}^s)\) for the family of all \(\delta \mathcal {S}\)-sets whenever X is a subspace of a strongly super weakly compactly generated space.